ASTRONOMY. 



[ACCELERATED MOTTO*. 



in the inn, describe* a curvilinear orbit round the 

 source of attraction, such that equal sectorial areas are 

 generated by the rariitu vector of the pUuict in equal time*. 

 However accordant this description of equal areas in 

 equal times may be with actual observation, we could 

 not fairly infer that the centre of force is necessarily in 

 the son, unless it were further proved that such equal 

 areas could not be described if the centre of attraction 

 were situated elsewhere than at the common vertex f 

 the equal sectors. And this may be done thus : If the 

 force acting on the planet when at B (see Fig. 144) had 

 a direction out of the line 13 S, B N would make some 

 angle with B S, and C M, which is parallel to B N, 

 would not then be parallel to 11 S ; and the two triangles 

 BUS, B M S, on the same base B S, would have their 

 vertices at unequal distances from that base ; so that 

 the surfaces of those triangles would be unequal. The 

 triangle A B S, always equal to B M S, would no longer 

 be equal to the triangle BCS; nor, consequently, 

 would the triangles ABS, BCS, CDS, *c. (Fig. 14o), 

 described by the radius vector in equal times, be equal. 

 \\ i- conclude, therefore, that, 1st, if the force acting on 

 a planet be constantly directed towards the sun, the 

 areas described by the straight line joining the planet 

 and sun are proportipnal to the times of describing 

 them ; and, 2nd, if the force which acts on the planet be 

 not directed towards the sun, the proportionality of the 

 sectorial areas to the times does not exist The first law 

 of Kepler, therefore, warrants the conclusion that the 

 attractive force acting on a planet, is constantly directed 

 towards the sun ; and this is the only conclusion deriv- 

 able from that law. As respects the intensity of the 

 attractive force how it varies with the distance of the 

 planet, or whether it varies at all, are particulars of 

 which nothing can be learnt from the observed fact of 

 the equal description of areas; for, as the preceding 

 reasoning shows, equal areas are always described about 

 a point, provided only the force acting on the moving 

 body be in the direction of the line from the body to the 

 point, however the intensity of the force may vary. But 

 if the body move in a circle round the centre of attrac- 

 tion, then we may infer, from the law of equal areas, in 

 equal times, that its motion must be uniform ; for only 

 equal arcs of circles can belong to equal sectors ; and 

 from the constancy of the distance, or radius vector, we 

 cannot but infer the constancy of the attractive force. 



Kepler's other two observed laws of planetary motion 

 evidently imply a certain law of attractive force. If the 

 attraction be such as to cause the planet to describe an 

 elliptic orbit, any alteration in the force acting upon the 

 planet at any particular point in that orbit, would 

 necessarily alter its path. And if two planets revolve 

 round a centre of force, and it be observed that their 

 mean distances from that centre and their times of revo- 

 lution bear the same relation to one another, a uniform 

 law of force is again implied. Let us endeavour to dis- 

 cover what this law is from Kepler's third proposition ; 

 namely, that the squares of the times are as the cubes of 

 the mean distances. 



It will be observed that the enunciation of this tmth 

 involves no condition in reference to the tccentricitiet 

 of the planets ; the proposition should hold, then 

 for all eccentricities, and even for circles, in which the 

 eccentricities are zero. For simplicity, then, we shall 



. consider the case of planets revolving in circular orbits, 

 and consequently, as shown above, moving uniformly ; 

 and shall thence endeavour to discover the law of force 



i necessary to justify the proportion, that the squares of 

 the times of revolution are as the cubes of the radii of 

 the circular orbits. 



The intensity with which a force acts npon a body, is 



\ measured by the addition it imparts to the body's velo- 

 city in tins direction of that force in a second of time. 

 H no force act, no addition can be made to velocity ; if 

 a body be at rest, or move uniformly in a straight line, 

 r at once that it is not acted upon by any force. 

 If a force act continuously, and in the same direction, it 

 Bust generate equal increments of velocity in equal 

 tunes ; and the motion of the body is thus said to be 



uniformly accelerated. It must be borne in mind, how- 

 ever, that the force must act with the same intensity 

 upon the body throughout Tim attraction of the earth, 

 or terrestrial gravity, may be regarded as such a con- 

 stant force in all our experiments and observations 

 respecting bodies falling to its surface, because these 

 observations and experiments can never be made in a 

 region so remote from the surface as to render the 

 variation in the force of the earth's attraction sensible. 



If a body fall from rest to the surface of the earth, it 

 will acquire a velocity at the end of the first second, 

 which, as stated above, is the measure of the earth's 

 attraction; throughout this second, the velocity in- 

 creases uniformly from nothing up to the final velocity 

 which measures the force. The velocity midway is, 

 therefore, half this final velocity ; and if the body were 

 to move uniformly with this midway velocity, it is 

 pretty obvious that in one second it wonld describe the 

 same length of path, as in falling from rest as above 

 supposed.* 



The length of path in nniform motion is evidently 

 found by multiplying the uniform velocity (the length 

 passed through in a second) by the number of seconds ; 

 that is, calling the length of space s, the velocity *, and 

 the number of seconds t, wo have =*, or if t 1, 

 j = . Now, if e be the final velocity at the end of the 

 second of accelerated motion, adverted to above, anil S 

 the space actually passed through in that second ; then, 

 putting $ V for r, and S for *, we have S = V ; there- 

 fore V = 2 S ; that is, the final velocity, or the measure 

 of the force of gravity, is twice the space described by a 

 body falling from rest in the first second of time. 



The same conclusion applies, however remote from the 

 surface of the earth the tailing body bo conceived to be 

 placed ; the intensity of gravity there would still be 

 measured by double the space through which the body 

 would fall from rest in one second ; for during so short a 

 time, the force may, as before, be considered as constant. 

 And a like conclusion applies whatever centre of force 

 be considered ; the measure of its intensity on a body 

 anywhere subjected to its action, will always be accu- 

 rately expressed by twice the space through which the 

 body falls towards that centre in one second. 



Let the circle in Fig. 146 represent the path of a 

 planet revolving about the sun at S. When at A, the 

 planet is actuated by a velocity in the direction of the 

 tangent A M, and in this direction it would, of course, 

 continue to move uniformly if no force diverted it from 

 its path. But it is acted upon by a force which is con- 

 stantly directed towards the sun, S, and which tends to 

 draw it towards that body, or to deflect it more and 

 more from the tangent. It may thus be said to fall 

 towards the sun, just as a body near the earth, projected 

 Fig. I. 



forwards in a straight lino, A M (Fig. 147), falls towards 



Bee itntf, Urthanict, p. 731, ft try. 



